# Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

## TL;DR

This paper develops high linear order implicit and implicit-explicit SSP Runge-Kutta methods that optimize stability properties for solving hyperbolic PDEs, extending the order beyond previous limitations for implicit methods.

## Contribution

It introduces new high linear order implicit and IMEX SSP Runge-Kutta methods with optimized stability regions, surpassing previous order restrictions for implicit SSP methods.

## Key findings

- Implicit SSP Runge-Kutta methods of any linear order were constructed.
- Optimized IMEX SSP methods with high linear and nonlinear orders were developed.
- Numerical tests confirmed the high order convergence and stability properties.

## Abstract

When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability coefficient. The search for high order strong stability time-stepping methods with high order and large allowable time-step had been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we aim to find very high linear order implicit SSP Runge-Kutta methods that are optimal in terms of allowable time-step. Next, we formulate an optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs that have high linear order and nonlinear orders p=2,3,4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.